est.SBA {graphon}

Estimate graphons based on Stochastic Blockmodel approximation

Description

est.SBA takes a 2-stage approach for estimating graphons based on exchangeable random graph models. First, it finds a Stochastic Blockmodel Approximation (SBA) of the graphon. Then, it uses clustering information to estimate graphon using a consistent histogram estimator.

Usage

est.SBA(A, delta = 0.5)

Arguments

A

either

Case 1.

an (n\times n) binary adjacency matrix, or

Case 2.

a vector containing multiple of (n\times n) binary adjacency matrices.

delta

a precision parameter larger than 0.

Value

a named list containing

H

a (K\times K) matrix fo 3D histogram.

P

an (n\times n) corresponding probability matrix.

B

a length-K list where each element is a vector of nodes/indices for each cluster.

References

Airoldi EM, Costa TB, Chan SH (2013). “Stochastic blockmodel approximation of a graphon: Theory and consistent estimation.” In Burges CJC, Bottou L, Welling M, Ghahramani Z, Weinberger KQ (eds.), Advances in Neural Information Processing Systems 26, 692–700. Curran Associates, Inc.

Chan SH, Airoldi EM (2014). “A Consistent Histogram Estimator for Exchangeable Graph Models.” In Proceedings of the 31st International Conference on International Conference on Machine Learning - Volume 32, ICML'14, I-208–I-216.

See Also

est.LG

Examples

## generate a graphon of type No.6 with 3 clusters
W = gmodel.preset(3,id=6)

## create a probability matrix for 100 nodes
graphW = gmodel.block(W,n=100)
P = graphW$P

## draw 17 observations from a given probability matrix
A = gmodel.P(P,rep=17)

## run SBA algorithm with different deltas (0.2,0.5,0.8)
res2 = est.SBA(A,delta=0.2)
res3 = est.SBA(A,delta=0.5)
res4 = est.SBA(A,delta=0.8)

## compare true probability matrix and estimated ones
opar = par(no.readonly=TRUE)
par(mfrow=c(2,2), pty="s")
image(P); title("original P")
image(res2$P); title("SBA with delta=0.2")
image(res3$P); title("SBA with delta=0.5")
image(res4$P); title("SBA with delta=0.8")
par(opar)